Quantitative Biology > Other Quantitative Biology

Abstract: Matrix forms of the representation of the multi-level system of
molecular-genetic alphabets have revealed algebraic properties of this system.
Families of genetic (4*4)- and (8*8)-matrices show unexpected connections of
the genetic system with Walsh functions and Hadamard matrices, which are known
in theory of noise-immunity coding, digital communication and digital
holography. Dyadic-shift decompositions of such genetic matrices lead to sets
of sparse matrices. Each of these sets is closed in relation to multiplication
and defines relevant algebra of hypercomplex numbers. It is shown that genetic
Hadamard matrices are identical to matrix representations of Hamilton
quaternions and its complexification in the case of unit coordinates. The
diversity of known dialects of the genetic code is analyzed from the viewpoint
of the genetic algebras. An algebraic analogy with Punnett squares for
inherited traits is shown. Our results are used in analyzing genetic phenomena.
The statement about existence of the geno-logic code in DNA and epigenetics on
the base of the spectral logic of systems of Boolean functions is put forward.
Our results show promising ways to develop algebraic-logical biology, in
particular, in connection with the logic holography on Walsh functions.